Developing Students’ through Games Mathematical Reasoning “D

Developing Students’
Mathematical Reasoning
through Games
“D
o you have a six?” Gina asked Martin.
“Yes, I do,” replied Martin as he
passed his card to Gina.
“So I have two cards that make ten,” said Gina
excitedly, showing her six of hearts and a four of
spades.
“Gina, do you have a five?” asked Carl.
“No,” replied Gina, and Carl drew a card.
will maximize their chance of winning. Engaging
mathematical games encourage students to explore
number combinations, place value, patterns, and
other important mathematical concepts. The three
Ps—plan, play, and please be patient—provide
a framework to help teachers consider a game’s
potential for exploring mathematical ideas with
students and leading to a rich discussion.
People of all ages love to play games. This
game, Make 10, was created for first-grade students
to help them find combinations of numbers with a
sum of 10 and to give them practice with number
facts (see fig. 1). The children work together to
make number pairs using a deck of cards. A team
“wins” if it makes the most pairs. Adults recognize that all the teams will have the same number
combinations (in other words, no group is going to
discover something unique), but this is not obvious
to children. Many times ideas that are obvious to
adults can launch a mathematical discussion in an
elementary school classroom. In this case, just ask
the children why the teams had the same number
combinations and why they think they all found
the same pairs. Good games for the classroom are
engaging and create opportunities for students to
explore mathematical ideas.
Kindergarteners and first graders enjoy games
based on chance. These games provide opportunities to explore fundamental number concepts
such as the counting sequence, one-to-one correspondence, and quantity. Third- and fourth-grade
students are intrigued by games of strategy, which
require players to consider multiple options, predict future moves, and plan a series of moves that
The Three P s
By Jo Clay Olson
Jo Clay Olson, [email protected], taught for twenty-five years at all levels, from preschool through
high school. She now teaches methods and content courses for prospective elementary teachers at
Washington State University, Pullman, WA 99164-2132.
464
When choosing a game for the classroom, first play
the game yourself with a family member or a colleague to gain familiarity with its rules and subtleties. Discuss the mathematical ideas embedded in
the game and how these ideas may emerge during
play. Determine the level of competition appropriate for your students and decide whether the rules
need to be modified to meet their needs. Anticipate
some possible responses or strategies that your
students may use while playing the game. From
your list of anticipated responses, create a list of
questions that you can ask to probe your students’
thinking during play.
Plan
Plan how you will introduce the game to your class.
Will two students demonstrate while you explain?
Will you invite one student to play the game with
you? Or will you play against the whole class?
Decide on an appropriate amount of class time to
devote to playing the game. Remember that playing a game for the first time requires a period of
learning and clarification. As students become
more familiar with the game, they will spend less
time learning the rules and more time exploring
mathematical ideas.
Decide how you will pair students or form the
small groups who then start the game on their
own. Students who use less sophisticated strategies
should be partnered with students who use slightly
Teaching Children Mathematics / May 2007
Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Photograph by Jo Clay Olson; all rights reserved
more mature strategies but not with the most
abstract thinkers. Abstract thinkers and students
who use immature strategies may be unable to
communicate with each other because they conceptualize mathematical ideas in very different ways
(Carpenter et al. 1999). The slightly more mature
thinkers can also be paired with the abstract thinkers because they use language with common meanings. As they explain their solutions or strategy, the
more mature thinkers develop their articulation of
mathematical ideas and thus lead less mature thinkers to more abstract ideas.
Play
Introduce the game and then, while the students
play, walk around the room and listen to their
conversations. Ask probing questions and listen
to the students’ responses. Rather than restate a
student’s strategy, ask his or her partner to explain
it. Take notes to record the different strategies that
your students use and to plan the class discussion.
Decide which strategies to discuss first, beginning
with less mature ideas and then moving to more
sophisticated strategies. Ask the students how the
Teaching Children Mathematics / May 2007
strategies are similar and how they are different.
Possible discussion starters include these:
• “Thumbs up if you liked the game, thumbs
sideways if it was okay, and thumbs down if
you didn’t like it. What did you like about it?
Why?”
• “What did you notice while playing the game?”
• “Did you make any choices while playing?”
• “Did anyone figure out a way to quickly find a
solution?”
After the discussion, ask the students to partner
with a different person to explain their strategy and
then give them five minutes to write the alternative
strategy in their mathematics journal. Last, suggest
that the next time they play the game they use a
different strategy.
Please be patient
Provide repeated opportunities for your students
to play the game, and let the mathematical ideas
emerge as they notice new patterns, relationships,
and strategies. Watch carefully how students incor465
Figure 1
Rules for playing the game Make 10
Make 10
Goal
Make number pairs with a sum of 10.
Players
Game is played by two or three children.
Rules
1. Shuffle a deck of cards with the numbers 0 to 10 or 1 to 9.
2. Deal five cards to each player. Place the remaining cards
face down on the table.
3. Player 1 asks one of the other players for a card to add
to one of her or his cards to make a sum of 10. The
requested card is then placed with a second card from
player 1’s hand, and the other players check the sum. If
the player does not have the requested card, player 1
draws one card from the face-down stack. If player 1 can
make a sum of 10 with two cards, the pair is placed on
the table.
4. The players draw additional cards from the face-down
stack until they each have five cards. If player 1 cannot
make a sum of 10 with the cards in her or his hand, player
1 keeps the six cards and does not draw additional cards
until he or she has fewer than five cards.
5. The game is over when the face-down cards have been
used up. The students count the number of pairs that
they made, and the group with the largest number wins.
(Note: Because the game should result in every group
finding the same number of pairs, everyone should win.
This outcome can prompt a rich discussion as to why this
is the case.)
Modifications 1. Play the game competitively; each player tries to get the
greatest number of pairs.
2. Allow students to use two or three cards to make a sum
of 10.
3. Change the goal from making 10 to creating the largest
two-digit number (this game is called Double Digits).
4. Change the goal from making 10 to making the highest
sum with two cards (this game is called Super Sums).
porate more abstract strategies into their own. Be
patient and allow the mathematical ideas to develop
over time. Your patience empowers students to
independently explore mathematical ideas and
create conceptual understandings that they will
not forget. To illustrate the three Ps framework
for using games in the classroom, three games are
presented here along with a brief discussion of
primary and intermediate students’ developmental
needs while playing games.
466
A Game for Primary-Level
Students: Close to 20
Games that five-year-olds like to play are generally based on luck: With each card a player draws
or with each roll of a die, the player has only one
choice. Board games, card games, and games with
dice allow primary-level students to explore fundamental number concepts by counting and using
one-to-one correspondence as they move a game
piece on a board (Fosnot and Dolk 2001). Eventually, the children notice that larger numbers in the
counting sequence yield a greater number of spaces
on the game board, and they begin to conceptualize
two aspects of numbers: Numbers represent both an
element in the counting sequence and a quantity.
First and second graders enjoy using mathematical reasoning to play more sophisticated games.
Competition should be minimized to keep their
thinking focused on the mathematical ideas. The
game Close to 20 (Akers et al. 1997) promotes
mathematical thinking while offering practice in
number facts. I was introduced to this game while
team teaching with a first-grade teacher. The different strategies that the children used to find number
combinations close to 20 intrigued me, and the
game quickly became a classroom favorite.
Close to 20 is played by using a set of cards
numbered 0 to 9 and recording sheets. Each player
is dealt five cards and then chooses three of the five
cards to make a sum as close to 20 as possible (but
not more than 20). A player’s score for the round
is the difference between his or her combination
and 20. On the recording sheet (fig. 2), each player
writes his or her combination, the total of the combination, and the score for the round (20 – total). At
the end of the round, each player discards the cards
used to make this first combination and receives
three new cards. Play continues for five rounds,
when the game is over. To minimize competition
and enhance cooperation, have each student pair
or student group aim to have the lowest combined
total in the class.
Plan
When I played this game, I quickly realized that I
was not finding all the possible sums close to 20. So
I tried some different strategies. One strategy I used
was to pick the largest card of the five I was dealt and
then pick two more cards whose sum was greater than
10. I knew that for the sum of my three cards to be
20, the sum of these two cards had to be equal to the
difference between my largest card and 10. Another
Teaching Children Mathematics / May 2007
strategy I used was based on my knowledge that 3
times 6 is 18 (a number close to 20 but still less than
20). Thus, I would select three cards that were close
to the number 6 (e.g., 6, 5, and 7).
The mathematical ideas underlying Close to 20
include these:
• Numbers can be combined in different ways.
• The closest sum may be greater than 20.
• Estimation and related facts can reduce the
number of combinations you have to check for
the closest sum.
During the second week of school, I wanted to
assess whether a group of second graders recognized that numbers could be combined in several
ways to make a larger number and to ascertain
their knowledge of number facts. Close to 20 was
a game I could use both as a formative assessment
and as a prompt for a mathematical discussion. I
decided to minimize competition by encouraging
the students to help one another find sums close to
20 and making the objective a lower score.
Play
To introduce the game, I played the game with
another student.
Teacher. We are going to play a card game called
Close to 20. What do you think that we’re going to
try to do?
Yasmeen. I know—we can use cards to make 20.
Teacher. Do you think that we’ll always be able
to make 20?
Marquis. Yes. You can make lots of numbers
with cards.
Larry. No. … Sometimes you get only little
cards and can’t make a big number.
Teacher. I wonder what you will find out. Who
wants to play the game with me? (Hands went up.)
Okay, Luis, come sit here. Please deal me five cards
and then give yourself five cards. Now, let’s look
at my cards. Who can help me make 20 with three
cards? (We used 5, 6, and 7 to make 18.) Now, here is
my recording sheet. What should we write down?
Marquis. Put the numbers on the lines and write
your sum here (pointing at the space next to the
equals symbol).
Teacher. What should we do here? (I pointed at
the last column.)
Shaundra. Put 2 in there, you need 2 more to
get to 20.
Teacher. Now let’s look at Luis’s cards. (Luis
shared his cards and possible combinations. Most
of the students were ready to play, and I gave cards
and two recording sheets to pairs of children.)
While the second graders played the game,
I walked around the room. I noticed that they
recorded the first combination that they created.
When I had demonstrated the game, several children suggested combinations, and we picked the
Figure 2
Recording sheet for the game Close to 20
Close to 20 ­Recording Sheet
Name ___________________________________
__________
Score for
Round
(20 – total)
__________
__________
__________
__________
__________
__________
__________
__________
__________
Total
__________
__________
__________
__________
+
+
+
+
+
__________
__________
__________
__________
__________
+
+
+
+
+
__________
__________
__________
__________
__________
=
=
=
=
=
Game Total __________
Teaching Children Mathematics / May 2007
467
Please be patient
Photograph by Jo Clay Olson; all rights reserved
The second graders played the game for 15 more
minutes. To construct sums that were closer to 20,
the children first selected the largest number in their
hand. They understood that different numbers could
be combined to make a sum of 20, but I did not see
them trying several sums before recording one. A
few children used base-ten blocks, multilink cubes,
and hundreds charts as tools to help them find combinations and the difference between their sum and
20. During my observation of their resumed play, I
saw children using some of the ideas that we had discussed. I wondered whether calculators would allow
them to find the combination that was the closest to
20. Would calculators enhance their exploration of
number patterns and relationships, or would they
become a tool for finding an answer? I realized that
providing calculators might change the focus of the
game and that I needed to carefully consider how
and when to introduce them.
largest one. Clearly, the students were focused on
finding only one combination that was less than
20. When I asked the children why they used a particular combination, I heard two responses: “I just
picked it” and “I knew that 4 plus 2 plus 5 is less
than 20 because they are all little numbers.” When
I asked how they figured out the difference between
their sum and 20, the students replied, “I counted
on my fingers.” After ten minutes of play, I decided
that it was time for us to discuss the game.
I began our discussion by asking, “Do you
remember when we talked about the game and we
wondered if we could make 20 every time? What
happened?” The second graders now voiced their
observation that it was hard to get 20. Then I asked,
“Did any of you make 20?” The students looked at
their sheets, and three children raised their hands.
I wrote their combinations on the board and asked
the class to look for a strategy that might help them
make combinations of 20. My questions and comments helped the second graders think more deeply
about different strategies for selecting cards:
• “What do you notice about these
combinations?”
• “Oh, they all begin with the largest number.
What do you think the children were thinking
about?”
• “How did you figure out sums of numbers without counting on your fingers?”
• “How could you use combinations that you
know to play the game?”
468
Cautions
Sometimes teachers use games solely to practice
number facts. These games usually do not engage
children for long because they are based on memorization. Some children are quick to memorize, while
others need a few moments to use a related fact to
compute. Children placed in situations in which
recall speed determines success may infer that being
“smart” in mathematics means getting the correct
answer quickly instead of valuing the process of
thinking. Consequently, they may feel incompetent
when they use number patterns or related facts to
arrive at a solution and may begin to dislike mathematics because they are not fast enough.
A Game for IntermediateLevel Students: The Product
Game
Students in the intermediate grades are intrigued
by games based on strategy and competition. All
strategy games provide students with opportunities
to make choices about the best strategy to use while
allowing them to explore mathematical ideas. Discussion about different strategies creates a context
for developing students’ conceptual understanding
of patterns, related number facts, and place value.
Many students use a peer’s explanation to support
their own learning by modifying it to one that makes
sense or comparing the strategies for the “best” one.
A reflective question such as “What did you think
about when you selected a number pair?” encourTeaching Children Mathematics / May 2007
ages students to articulate their mathematical ideas.
Following is a discussion of the Product game to
illustrate the three Ps.
Plan
In the Product game, students use a variety of strategies to decide the best first move. The goal of the
game is to claim any four squares in a row or in a
column, any four squares that form a diagonal, or the
four squares at the corners. Figure 3 shows a game
board designed for fourth-grade students to explore
the products that can be made from the factors 0, 1,
2, 3, 4, 5, and 6.
To begin, player 1 places a paper clip on one
factor on the game board. Player 2 places a second
paper clip on another factor and claims the product
of these two factors by placing a colored chip on
that square on the game board. Player 1 then moves
one of the clips to a different factor and claims the
resulting product. The players continue taking turns
by moving one paper clip and claiming the resulting
product until one player wins—that is, claims four
squares—or a draw is declared.
The first time I played the Product game with
the factors 0 through 6, I realized that in my mind
I was substituting other factor pairs for the products
on the game board. I was amazed at the number of
facts that I quickly considered while deciding which
product to claim. Later, I noted that the factors 0, 1,
and 2 were used to generate more products on the
game board than the factors 4, 5, and 6 and decided
that it was better to claim products smaller than 16.
I claimed the larger products when the opportunity
arose and when they extended a row, column, or
diagonal.
The mathematical ideas underlying the Product
game include these:
• A product is generated by a pair of factors.
• Some products can be created by more than one
factor pair.
• The game board changes when different factors
are used.
• A finite set of factors can be combined to generate a finite number of products.
Play
I usually introduce this game by playing against the
class. After briefly explaining the rules, I place one
paper clip on a factor on the game board and then
select a student to pick another factor. The class
claims the resulting square with a colored chip. I
move one paper clip to another factor and place a
Teaching Children Mathematics / May 2007
different colored chip on my square. After several
moves, I win and challenge the class to one more
game. After the second game, the students are ready
to pair up and play against each other. While the student pairs play, I engage students in conversations.
Following is an excerpt from a discussion that I had
with two fourth-grade students.
Teacher. (I placed a marker on 5, and José
claimed the square with the number 10 by placing
his marker on 2.) José, that is an interesting first
move. Why did you pick it?
José. There are five ways to win.
Teacher. Cherie, do you see any of the ways that
José could win?
Cherie. Well, he could win by making a column,
row, or either diagonal. That’s only four ways.
José. There are two ways on the column. I could
use 5, 10, 16, and 25 or 1, 5, 10, and 16.
Teacher. Interesting. Cherie, what do you think
is the best first move?
Cherie. I like to take 5 first.
Teacher. Why?
Cherie. It’s easier to get four in a row.
Teacher. Why is it easier?
Cherie. Uhm ...Well, the numbers are smaller
and that makes it easier.
Teacher. José, what do you think?
José. There’s only three ways that you can, so it
isn’t as good.
Cherie. But some of the products can be gotten
in lots of ways so it’s easier to fill them in.
José. What do you mean?
Cherie. See, you can get 4 by either 1 and 4 or
putting both clips on the 2. Six is the same way, so
it’s easier to win.
Teacher. Those are both great strategies. Why
don’t you try to find out which one is better?
Figure 3
Game board for the Product game for use
with the factors 0 through 6
0
4
9
15
24
1
5
10
16
25
2
6
Free
18
30
3
8
12
20
36
469
Figure 4
Student-constructed 6-by-6 game board for the Product game for use
with the factors 1 through 9
1
2
3
4
5
6
7
8
9
10
12
14
15
16
18
20
21
24
25
27
28
30
32
35
36
40
42
45
48
49
54
56
63
64
72
81
In this dialogue, the reflective questions I asked
prompted José and Cherie to explore two different
mathematical ideas. José explored the patterns of
rows, columns, and diagonals to determine the best
first move. Cherie considered the number of factor
pairs that would enable her to fill in the squares.
Both used mathematical reasoning to predict possible
outcomes for their strategies. Initially I interpreted
Cherie’s response that she liked the smaller number
to indicate knowledge of the products using the factors 1, 2, and 4, but it became clear that she based her
strategy on the relationship between factor combinations and products. I encouraged the two students
to explore each other’s reasoning by investigating
which strategy was “better.” Investigations based on
students’ mathematical thinking prompt mathematical discussions among students as they compare the
strengths and limitations of different strategies.
Please be patient
These fourth graders enjoyed playing the Product
game for many months. We extended the game by
using the factors 1 through 9 and created a 6-by6 game board (see fig. 4). The students designed
a double-elimination tournament conducted during indoor recess and after school. Students who
were eliminated during the tournament watched
their peers play against one another, talked about
strategies, and designed a round-robin tournament
for themselves. I found myself listening to their
discussions, impressed with their articulation of
mathematical ideas, development of computational
fluency, and enthusiasm for learning and using
mathematics. Learning was enhanced when I cre470
ated opportunities for students to explore, reflect,
and discuss their mathematical observations by providing time, asking questions, and being patient.
Cautions
Many students enjoy playing games in which the
person with the quickest correct response wins and
continues to play against other class members—for
example, Around the World or Math Baseball.
Games of this type pose an ethical question, however: Does the game provide an equal opportunity
for all students to gain fluency with number facts?
Students who need the most practice with number
facts usually sit at their desks while the students
who know the facts have more opportunities to
practice. Thus, the very students who need to practice have less opportunity to learn. When struggling
students have a turn, they do not have time to use
mathematical reasoning and often simply guess.
Often their responses are incorrect, and they return
to their seat, silently hoping to avoid another turn.
Summary
Games are fun and create a context for developing
students’ mathematical reasoning. They provide
opportunities for students to wonder why some
peers are quick to respond and thus encourage
students to compare different strategies. Through
playing and analyzing games, students also gain
computational fluency by describing more efficient strategies and discussing relationships among
numbers. Teachers can create opportunities for
students to explore mathematical ideas by planning
questions that prompt students to reflect about their
reasoning and make predictions.
Driscoll (1999) suggests that teachers sometimes limit their questions to three types: managing, clarifying, and orienting. Managing questions
help students focus on the problem and begin to
work. Clarifying questions help students interpret
the problem and select a problem-solving strategy
that will lead to a correct solution. Orienting questions keep students thinking about the problem
and motivate them toward a correct answer.
To develop students’ conceptual understanding,
Driscoll encourages teachers to expand their repertoire of questions to include those that prompt
reflection and those that elicit algebraic reasoning.
While the children were playing Close to 20, I
prompted their reflection by asking why they used
a particular combination and how they figured out
the difference between their sum and 20. Using
Teaching Children Mathematics / May 2007
their responses, I planned a discussion that would
encourage them to reflect on whether they could
always generate a sum of 20. Then I encouraged
them to use algebraic reasoning to find a numeric
pattern when comparing three solutions that generated a sum of 20. These questions prompted the
students to use more sophisticated thinking to generate sums that were closer to 20.
Teachers help students develop algebraic reasoning by asking questions that prompt them to
solve problems by using forward and backward
processes (Driscoll 1999). For example, in the
Product game I prompted forward thinking by
asking, “What are the best first moves? Why?”
Such questions encourage students to think several
moves ahead and identify factors that will generate
products that will have them well positioned on the
game board. I encouraged backward thinking by
asking students to begin with a product and then
identify factors that formed the given product. For
this game, a good backward-thinking question is,
“What if our opponent could win by claiming the
square with a 16. What factors would we want to
avoid placing a paper clip on?”
Teaching Children Mathematics / May 2007
As students play games and analyze strategies,
they explore mathematical ideas and compare different strategies for efficiency. The three Ps format
can help teachers use games to develop students’
conceptual understanding. When we carefully consider the questions we ask and plan an appropriate
level of competition, students stay focused on the
mathematics instead of on winning.
References
Akers, Joan, Michael Battista, Mary Berle-Carman, Douglas Clements, Karen Economopoulos, Ricardo Nemirousky, Andee Rubin, Susan Russell, Cornelia Tierney,
and Amy Weinberg. Investigations in ­Number, Data,
and Space. Palo Alto, CA: Dale ­Seymour, 1997.
Carpenter, Thomas, Elizabeth Fennema, Megan Franke,
Linda Levi, and Susan Empson. Children’s Math­
ematics: Cognitively Guided Instruction. Portsmouth,
NH: Heinemann, 1999.
Driscoll, Mark. Fostering Algebraic Thinking: A Guide
for Teachers Grades 6–10. Portsmouth, NH: Heinemann, 1999.
Fosnot, Catherine, and Maarten Dolk. Young Mathema­
ticians at Work: Constructing Number Sense, Addi­
tion, and Subtraction. Portsmouth, NH: Heinemann,
2001.s
Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
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